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Compressed Sensing for Diffusion MRI of the Mouse Brain:A Trade-off of Spatial and Directional Information
by
Anthony Albert Salerno
A thesis submitted in conformity with the requirementsfor the degree of Master of ScienceDepartment of Medical Biophysics
University of Toronto
c© Copyright 2018 by Anthony Albert Salerno
Abstract
Compressed Sensing for Diffusion MRI of the Mouse Brain: A Trade-off of Spatial and
Directional Information
Anthony Albert Salerno
Master of Science
Department of Medical Biophysics
University of Toronto
2018
Diffusion Magnetic Resonance Imaging (MRI) uses loss of signal coherence caused by
movement of water molecules to determine how water diffuses in tissue. These scans are
known to take a significant amount of time owing to the repeated acquisition of images,
each measuring diffusion for a particular direction and weighting. Emerging methods
based on collection of increasingly large numbers of diffusion directions and/or weight-
ings promise to provide a more detailed characterization of brain microstructure than
previously possible. However, because the time required for these methods can render
them impractical, I present a method of acquiring and reconstructing diffusion MRI data
based on Compressed Sensing (CS) that enables flexible trade-off between spatial sam-
pling and diffusion direction/weighting sampling. This method enables the collection of
more diffusion data in a given scan time at the cost of extended reconstruction time.
ii
This work is dedicated to my late cousin Enzo Cardillo who spent many days and
nights at The Hospital for Sick Children during his short life. No matter what state he
was in, he did everything he could to make me smile. It didn’t matter that he barely
had the energy to go to school, whenever he saw me, he made me laugh and be able to
just enjoy being a kid. If it wasn’t for him, I doubt I would have had the passion I do
for my work, or the gratitude for the amazing research that occurs at The Hospital for
Sick Children. In the same breath, I want to thank the doctors, nurses, and staff who
worked tirelessly to give me a few extra years with him.
Acknowledgements
Many thanks go to my parents Mary and Frank Salerno, my siblings Stefania, Muriel,
and Cesare, as well as Elisa De Luca for their love and support during this process, I
know I wasn’t the easiest person to deal with. I also want to thank all of the staff and
students at the Mouse Imaging Centre for their continued support and chats at my desk
where many coffees were shared.
Thank you to my committee members Dr. John Sled and Dr. Chris Macgowan for their
words of encouragement, probing questions, and brilliant ideas during my committee
meetings.
And finally, a special thank you to my supervisor Dr. Brian Nieman, who put in long
hours to help me with my project and to keep the Mouse Imaging Centre running at its
full potential.
iii
Contents
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction 1
1.1 Magnetic Resonance Imaging . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 The Signal of MRI . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Diffusion-Weighted Magnetic Resonance Imaging . . . . . . . . . . . . . 4
1.2.1 The Diffusion Process . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Diffusion MRI Sequence . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.3 Diffusion Tensor Imaging . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Higher Order Models of Diffusion MRI . . . . . . . . . . . . . . . 10
Diffusion Kurtosis Imaging (DKI) . . . . . . . . . . . . . . . . . . 10
High Angular Resolution Diffusion Imaging (HARDI) . . . . . . . 11
Neurite Orientation Dispersion and Density Imaging (NODDI) . . 11
1.2.5 Clinical Uses of Diffusion MRI . . . . . . . . . . . . . . . . . . . . 12
1.3 The Mouse as a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.1 MRI in the Mouse . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.2 Diffusion MRI in the Mouse . . . . . . . . . . . . . . . . . . . . . 14
1.4 Compressed Sensing (CS) . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4.1 Introduction to Compressed Sensing . . . . . . . . . . . . . . . . 15
1.4.2 Current Uses of Compressed Sensing in Human Research . . . . . 19
iv
1.5 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2 Methods 22
2.1 Adaptation of Compressed Sensing for Diffusion Data . . . . . . . . . . . 22
2.1.1 Total Variation in the Diffusion Dimension . . . . . . . . . . . . . 23
2.1.2 Mathematical Approximations and Definitions . . . . . . . . . . . 24
`1 Norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
TV Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1.3 Phase in CS Reconstructions . . . . . . . . . . . . . . . . . . . . . 26
2.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Choosing Directions for Diffusion Acquisition . . . . . . . . . . . 28
2.2.2 Spatial Undersampling Across Diffusion Directions . . . . . . . . 29
2.3 Implementation of ex vivo Diffusion RARE . . . . . . . . . . . . . . . . . 32
2.3.1 Acquisition Parameters . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.2 Phase Encode Partitioning Scheme . . . . . . . . . . . . . . . . . 33
2.3.3 Correction of Diffusion-Gradient Induced Phase . . . . . . . . . . 34
Interleaving ±~d Acquisitions: The “Checkerboard” . . . . . . . . 34
Diffusion-Gradient Induced Phase Corrections . . . . . . . . . . . 36
2.4 CS Reconstruction with ex vivo Diffusion RARE . . . . . . . . . . . . . . 38
2.4.1 Slice-to-Slice Correction . . . . . . . . . . . . . . . . . . . . . . . 38
2.5 Mice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3 Results 43
3.1 CS Reconstruction – Proof of Principle . . . . . . . . . . . . . . . . . . . 43
3.1.1 Retrospectively Undersampled Shepp-Logan Phantom . . . . . . . 43
3.1.2 Retrospectively Undersampled Anatomical in vivo Data . . . . . . 43
3.2 Diffusion MRI Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 “Checkerboard” Correction of Diffusion-Gradient Induced Phase . 45
3.2.2 Compressed Sensing Reconstruction . . . . . . . . . . . . . . . . . 47
3.3 Potential Use for HARDI Analysis . . . . . . . . . . . . . . . . . . . . . . 49
v
4 Discussion 55
4.1 Phase Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.1.1 ~d and -~d “Checkerboard” Acquisition Scheme . . . . . . . . . . . 56
4.2 Limitations of the Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.1 Phase Measurements . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2.2 Trade-Offs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.2.3 Empirical Tuning of the Weighting Factors (λi) . . . . . . . . . . 57
4.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.1 Further Applications in Diffusion . . . . . . . . . . . . . . . . . . 58
4.3.2 Potential Use in Clinic . . . . . . . . . . . . . . . . . . . . . . . . 59
5 Conclusion 60
Bibliography 62
vi
List of Figures
1.1 RARE pulse sequence example with the use of a diffusion gradient . . . . 7
1.2 Sparsifying terms applied to a Shepp-Logan phantom . . . . . . . . . . . 17
2.1 Comparison of 1a
ln(cosh(ax)) and sgn(x) . . . . . . . . . . . . . . . . . . 26
2.2 Differences in the probability density functions for uniform and variable
density undersampling methods . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Uniform density undersampling method with directional biases included . 32
2.4 Checkerboarded sampling pattern with ±~d . . . . . . . . . . . . . . . . . 35
2.5 Flowchart of phase corrections . . . . . . . . . . . . . . . . . . . . . . . . 39
2.6 Simplified flowchart of the data collection and processing using a diffusion
specific CS acquisition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7 Flowchart of the CS algorithm for a diffusion acquisition . . . . . . . . . 41
2.8 Slice-to-slice intensity corrections . . . . . . . . . . . . . . . . . . . . . . 42
3.1 Shepp-Logan phantom CS reconstruction . . . . . . . . . . . . . . . . . . 44
3.2 Anatomical CS reconstruction . . . . . . . . . . . . . . . . . . . . . . . . 45
3.3 Representative phase corrections . . . . . . . . . . . . . . . . . . . . . . . 46
3.4 Comparison of fully-sampled and undersampled image reconstructions for
an individual diffusion direction. . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Comparison of fully-sampled and undersampled image acquisitions with
matched acquisition times. . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.6 Comparison of FA maps in 30-direction fully-sampled and 120-direction
undersampled diffusion acquisitions. . . . . . . . . . . . . . . . . . . . . . 51
3.7 A depiction of the complete 30-direction and 120-direction data sets. . . 52
vii
3.8 HARDI analysis of 120-direction undersampled diffusion MRI data. . . . 54
viii
List of Abbreviations
ADC Apparent Diffusion Coefficient.
CEST Chemical Exchange Saturation Transfer.
CS Compressed Sensing.
CSF cerebro-spinal fluid.
DKI Diffusion Kurtosis Imaging.
DSI Diffusion Spectrum Imaging.
DTI Diffusion Tensor Imaging.
EPI Echo Planar Imaging.
FA Fractional Anisotropy.
fMRI functional MRI.
FOV field of view.
GDiff diffusion gradient.
GPE1 phase encode gradient 1.
GPE2 phase encode gradient 2.
GRO readout gradient.
ix
HARDI High Angular Resolution Diffusion Imaging.
MD Mean Diffusivity.
MRI Magnetic Resonance Imaging.
NODDI Neurite Orientation Dispersion and Density Imaging.
ODF orientation distribution function.
PDF probability density function.
QBI Q-Ball Imaging.
RARE Rapid Acquisition with Relaxation Enhancement.
RF radio frequency.
SENSE Sensitivity Encoding.
SNR signal-to-noise ratio.
TV Total Variation.
x
Chapter 1
Introduction
1.1 Magnetic Resonance Imaging
MRI plays an integral role in medical imaging in both the clinical and research worlds for
non-invasive anatomical, physiological, and functional measurements of the body. MRI
has the ability to acquire images with multiple different contrasts by changing the pa-
rameters of acquisition [1]. Having this flexibility is important for research and allows
the scientist the opportunity to optimize image acquisitions for the assessment of human
or preclinical models, in vivo or ex vivo samples, and for variability in tumour charac-
teristics, for instance [2, 3, 4, 5]. In biomedical research, MRI provides high anatomical
resolution with excellent soft tissue contrast; however, it is plagued by long scan times,
especially for methods that probe physiology or microstructure of the sample. The long
scan times for the latter methods are attributable to models of underlying microstructure
that are based on contrast changes in an image series, where the small intensity differ-
ences between each scan are used to help derive outcomes of interest. In the interest of
using MRI resources to obtain as much information as possible, it is important that im-
age acquisitions are tailored to balance spatial resolution and information for modelling
of microstructure within the time and resources available.
1
2
1.1.1 The Signal of MRI
In order to generate signal in MRI, first a strong static field ~B0 must be applied uniformly
over the sample. The following is described in a manner similar to [1].
Conventionally, ~B0 ≡ B0k. This generates a bulk magnetization ~M0 in the sample,
aligned with this strong field. In order to obtain signal, the net magnetization must
be rotated to have a component perpendicular to ~B0, which is achieved by applying a
radio frequency (RF) pulse (via another magnetic field, ~B1) at the Larmor frequency,
ω0 = γ| ~B0|, where γ represents the gyromagnetic ratio. Depending on the frequency
profile and duration of ~B1, ~M can be rotated to arbitrary angles from the main magnetic
field ~B0.
Once the net magnetization ~M has a component perpendicular to ~B0, it precesses
around ~B0. The precession can be detected through current induced in a RF coil via
the Faraday-Lenz law of induction, which is the signal that is measured. The precession
is described by both a magnitude and phase, represented through real and imaginary
components in the signal. The precessing magnetization is transient, and returns to its
equilibrium state as described by two relaxation times, T1 and T2, which represent the
longitudinal and transverse relaxation times respectively. The former (T1) describes the
rate of signal recovery along ~B0, while the latter (T2) describes the rate of signal decay
perpendicular to ~B0. This can all be represented by the Bloch Equation:
d ~M(t)
dt= ~M(t)× γ ~B(t)− (Mx(t)i+My(t)j)
T2
− (Mz(t)−M0)k
T1
(1.1)
~M(t) =
Mx
My
Mz
=
e−tT2 0 0
0 e−tT2 0
0 0 e−tT1
~Rz(ω0t) ~M0 +
0
0
M0(1− e−tT1 )
(1.2)
Where ~B is the total magnetic field and ~Rz is the rotation matrix for rotation about the
z-axis. It is also convenient to define Mxy such that the transverse component can be
3
written more simply.
Mxy = Mx + iMy (1.3)
dMxy
dt=dMx
dt+ i
dMy
dt(1.4)
dMxy
dt= −
(1
T2
+ iω0
)Mxy (1.5)
The solution to Equation (1.5) describes the evolution of transverse magnetization with
explicit relaxing and precessing parts:
Mxy = M⊥e−t/T2e−iω0t (1.6)
Where M⊥ is the magnitude of the transverse magnetization.
For imaging purposes, signal across space is distinguished by changing the magnetic
field strength linearly as a function of position. Detection of the associated change in
frequency (“frequency encoding”) or equivalently the change in phase over a defined unit
of time (“phase encoding”) thus maps signal to spatial coordinates. The imposed linear
change in magnetic field is generally referred to as a magnetic field gradient (henceforth
referred to simply as “gradient”). For each of the three dimensions, the gradients are
applied such that:
~B = ~B0 + ~G · ~r (1.7)
Where ~r represents the location relative to the isocentre of the magnet where the strength
of ~B0 is unperturbed. Incorporating the spatial variation induced by the gradients into
Equation (1.6), we may write:
Mxy(~r, t) = M⊥(~r)e−t/T2(~r)e−iω0te−iγ∫ t0~G(τ)·~rdτ (1.8)
Finally, to obtain signal off of this magnetization, we would integrate over all three
dimensions:
s(t) =
∫x
∫y
∫z
Mxy(~r, t) dx dy dz (1.9)
4
1.2 Diffusion-Weighted Magnetic Resonance Imag-
ing
1.2.1 The Diffusion Process
The process of molecular diffusion is defined by Brownian motion of particles as a result
of the thermal energy they hold [6]. In free space, the molecular displacement distribution
follows a uniform Gaussian in three dimensions; this is statistically described as a diffusion
coefficient, D. The diffusion coefficient is in SI units of m2s−1 (or more conveniently,
mm2s−1), relating to a mean squared displacement of each individual particle; for free
water at 37◦C this value is 3×10−3 mm2s−1 [7]. Within the brain, however, the molecular
displacements differ from an isotropic Gaussian model due to the presence of tissue
components such as cell membranes and macromolecules [7]. Within very short times in
these samples (short enough such that there will be minimal interaction with barriers),
the diffusion will look Gaussian. However as the diffusion time increases, the obstacles
become important and affect the diffusion observed.
1.2.2 Diffusion MRI Sequence
The microstructure of the brain can be probed by measuring how water diffuses in dif-
ferent directions or areas of the brain. Deviation from isotropic Gaussian diffusion is
particularly evident in white matter, as myelinated axons act as tracts, inhibiting the
diffusion of water perpendicular to the orientation of the axon. Diffusion MRI exploits
this phenomenon to elucidate the microstructure of the brain by probing the diffusion
rate of water within tissue at a multitude of different directions, ~d, with the expectation
that certain directions will restrict the diffusion of water more than others based on the
orientation of these barriers [8, 9, 10, 11, 12]. This indicates, for instance, the degree of
alignment of axon bundles.
Diffusion MRI is similar in its acquisition to anatomical MRI except for its one defining
factor – the introduction of strong diffusion sensitizing gradients. These gradients are
identical to the imaging gradients described, except applied with larger magnitude and
5
usually separated in time. They are applied within the pulse sequence in order to probe
diffusion of water, a single vector direction at a time, mapping the result over all space
in the image volume [13, 14, 15, 16, 17].
The detection of diffusion through MRI originates from protons that move along the
diffusion gradient. The notion is that where diffusion exists, signal is attenuated due to
a loss of phase coherence between the protons within a voxel. The method is as follows:
A diffusion gradient is applied, and generates a change in the phase of each proton as a
function of position along the sample in the direction of the gradient. After this gradient
is turned off, a 180◦ pulse is then applied (thus reversing the accumulated phase), and
the same gradient is applied again. For a stationary signal the amount of phase gained
from the first application of the diffusion gradient will be exactly cancelled by the second
gradient, effectively leaving zero net phase. However, if the proton has moved along the
probed axis, phase induced by the second gradient does not match the first, and a net
residual phase is present. This will lead to reduced phase coherence within the voxel and
a loss of signal. Increased levels of diffusion, through pathology, temperature or other
changes, lead to greater signal loss due to decreased phase coherence within each voxel.
Larger diffusion gradients and timings also produce larger phase deviations, and hence
greater decreases in signal intensity.
The impact of diffusion can be calculated beginning from Equation (1.6), but collect-
ing the T2 relaxation term and the precession term into a single value, S0, for simplicity
[18], the magnetization can be written as:
Mxy(~r, t) = S0e−iγ( ~G(t)·~r)t (1.10)
If diffusion within the sample follows Fick’s law, the magnetization of the system can be
described by a modified differential equation accounting for diffusion:
dMxy(~r, t)
dt=[−iγ(~G(t) · ~r) + D∇2
]Mxy(~r, t) (1.11)
Where ∇2 represents the Laplacian and D represents the diffusivity of the sample, which
here has been assumed to be isotropic. The solution to this differential equation can
6
be isolated by assuming a time-varying amplitude (referenced to S0) multiplied by a
precession term. It is convenient in this solution to define a parameter known as the
b-value – in units of s/mm2 – as follows:
Mxy(~r, t) = S0e−bDe−iγ( ~G(t)·~r)t (1.12)
b = γ2
∫ τ
0
[∫ t
0
G(t′)dt′]2
dt (1.13)
All diffusion imaging requires a comparison of data across two b-values, but the first
is generally set to b = 0, which is equivalent to a purely anatomical scan and called
the b0 image. For diffusion weighting based on a pair of matched idealized rectangular
gradients, as was solved in [18], the b-value can be simplified as follows:
b = γ2G2δ2
(∆− δ
3
)(1.14)
Where γ is the gyromagnetic ratio in MHz/T, G is the strength of the diffusion gradient
pulse in units of T/mm, δ is the duration of the pulse in units of ms, and ∆ is the time,
also in units of ms, between the beginning of the diffusion gradients. An example of
diffusion gradients (GDiff) and the respective timings are pictured as part of an imaging
sequence in Figure 1.2.
In order to probe the directional dependence of diffusion, an image is collected for
many different diffusion gradient directions covering the 3-dimensional Euclidian sphere,
which for a given b value will maintain a constant |GDiff|, but different combinations of
GRO, GPE1, and GPE2 to probe direction. The diffusion in any given region can be either
isotropic (not directionally dependent) or anisotropic (directionally dependent). If there
is isotropic diffusion at a given voxel, for all diffusion gradient directions there will be
consistent intensity differences from the b0 scan [19]. However, for anisotropic diffusion,
some directions will have higher intensities than others due to the restriction of water
diffusion. An example of this would be in the aforementioned white matter tracts where
myelin sheaths and axonal membranes act as barriers to motion perpendicular to their
principle axes, but have little effect on travel along their axis [19, 20, 21, 22].
7
Figure 1.1: One TR of a Rapid Acquisition with Relaxation Enhancement (RARE) sequence witha diffusion sensitivity gradient applied. The diffusion gradient (GDiff) is applied across the readoutgradient (GRO), phase encode gradient 1 (GPE1), and phase encode gradient 2 (GPE2) but is shownseparately for readability. The values of G, δ, and ∆ can be changed to obtain the desired b-value.RDG represents the read dephase gradient (in black), SG represents the spoiler gradient (in gray), RGrepresents the read gradient (in green), and PEG represents the phase encode gradient (unshaded).
Since an entire image must be acquired for each diffusion direction assessed, diffusion
MRI scans are known to take significantly longer than standard anatomical scans. Con-
sequently, other means of speeding image acquisition are usually employed including the
use of different pulse sequences, such as Echo Planar Imaging (EPI) [23, 24, 25].
The ability to infer microstructural changes from diffusion scans is dependent upon the
model employed to describe diffusion-induced intensity changes, which in turn determines
the number of directions and b-values acquired. More sophisticated, flexible models
require a large number of directions and/or b-values to probe finer angular dependence of
diffusion or diffusion restrictions. The simplest and most widely employed model is called
Diffusion Tensor Imaging (DTI) and will be discussed below along with a description of
more advanced models.
8
1.2.3 Diffusion Tensor Imaging
As one of the first models to explain diffusion MRI, DTI [21, 22, 26] attempts to probe
the anisotropy of water diffusion by generating an ellipsoidal probability distribution
[18, 27, 28, 29]. The model is applied across all voxels in a set of images and does not
use any prior information about anatomical location.
Through DTI, the intensity of images as a function of direction is fit according to the
diffusion tensor D.
Sk = S0 e−b gTk D gk (1.15)
D =
Dxx Dxy Dxz
Dyx Dyy Dyz
Dzx Dzy Dzz
(1.16)
In this representation, S0 represents the signal intensity of the b0 image, Sk is defined
as the signal intensity of the image at the kth diffusion direction, gk is a 3 × 1 unit length
column vector defining the kth diffusion gradient and D represents the diffusion tensor as
a 3×3 matrix [8, 21, 22]. By definition, the tensor is a symmetric matrix (i.e. Dij = Dji)
and as such there are only six unique elements that must be calculated – namely the
upper triangle. From the diffusion tensor, a number of diffusion properties can be inferred,
mainly based on the eigenvalues and eigenvectors of the diffusion tensor matrix. Here, it is
convenient to let λi represent the eigenvalues, and ~vi represent the individual eigenvectors.
Generally, the eigenvalues are presented such that λ1 ≥ λ2 ≥ λ3 ≥ 0. λ1 represents the
largest of the eigenvalues, and is often called the axial or parallel diffusivity – it can also
be represented by λ‖. Along with its eigenvector, it represents the degree and direction of
the greatest diffusion in the voxel. The other two eigenvalues can be averaged in order to
create the radial or perpendicular diffusivity, λ⊥, representing the diffusion in the plane
perpendicular to the axial direction. Finally, the Mean Diffusivity (MD) is the average of
all three eigenvalues – λ – which is also called the Apparent Diffusion Coefficient (ADC).
It is important to note that this measure is not sensitive to anisotropy, but will indicate
how much diffusion is occurring (an important indicator in many clinical applications,
9
such as detecting edema). In white matter, anisotropy occurs predominantly due to an
inhibition of diffusion in the direction of eigenvectors ~v2 and ~v3, with limited or no change
in the direction of ~v1. As such, the ADC is generally larger in voxels where λ1 ≈ λ2 ≈ λ3
where diffusion becomes less restricted, which can be an indication of pathological change
in white matter [21, 22, 26, 30, 31, 32, 33].
λ‖ = λ1 (1.17)
λ⊥ =λ2 + λ3
2(1.18)
λ = ADC =λ1 + λ2 + λ3
3(1.19)
In healthy white matter, or other areas of restricted diffusion, one can use the eigenval-
ues to define the degree of anisotropy within each voxel, and thus within different regions
in the tissue. The primary method of calculating anisotropy within the tensor model
is by a metric called Fractional Anisotropy (FA). The FA is a value in the range from
zero to one, where an FA of zero represents a perfectly isotropic voxel (λ1 = λ2 = λ3),
and an FA of one represents a voxel where diffusion can only occur in one direction
(λ1 > 0, λ2 = λ3 = 0). FA is calculated with the following equation:
FA =
√1
2
(λ1 − λ2)2 + (λ1 − λ3)2 + (λ2 − λ3)2
(λ21 + λ2
2 + λ23)
(1.20)
In an idealized voxel of white matter, diffusion will occur preferentially in one direction
so that λ1 � λ2 ≈ λ3 – describing an ellipsoid with one long and two equivalent shorter
axes with FA . 1. Another important case is when λ1 ≈ λ2 ≈ λ3, which is found
generally in grey matter and CSF. In some cases, the isotropic diffusion may be indicative
of pathology such as a tumour [32]. The eigenvalue information thus provides a useful
tool to indicate the degree of directional-specific diffusion occurring through FA maps,
widely used both in research and the clinic.
Through the use of eigenvectors, these maps can be made more informative by defining
the principle direction of diffusion at each voxel – generally by assigning a colour to each
principle direction and mixing the colours at each voxel based on the principle direction of
10
diffusion determined from ~v1. Combined, the directional information is given by colour,
and the intensity set by FA represents the preference for diffusion in that direction. This
representation can be used quite effectively to find brain tumours via delineation of white
matter surrounding them, and may be useful in surgical planning to determine how white
matter tracts have been distorted around tumour tissue [34, 35, 36, 37].
1.2.4 Higher Order Models of Diffusion MRI
Despite significant strengths of DTI, many regions of tissue are poorly described by el-
lipsoidal or isotropic diffusion. A classic problem case is two crossing fibres, such as
where white matter tracts cross one another. Here, diffusion in a single voxel contains
a mixture of unrestricted diffusion in two directions, and restricted diffusion in one, and
thus cannot be accurately described by a simple 3× 3 tensor. This leads to a relatively
small FA, even though there is significant anisotropy present at the microstructural level.
Moreover, the DTI framework does not model restrictions to diffusion per se, but ac-
counts for them empirically by allowing reduced diffusion in particular directions. This
motivated the development of alternative models that can describe more complicated
patterns of diffusion, with increased dependence on direction or degree of diffusion. This
in turn, depending on the model, requires acquisition of images for a larger set of b-values
and/or number of directions. Of course, in order to obtain these extra image contrasts,
additional scan time is necessary.
Diffusion Kurtosis Imaging (DKI)
Diffusion Kurtosis Imaging (DKI) advances the simple DTI model by characterizing the
deviation of diffusion from standard Gaussian diffusion, which is to be expected in regions
where membranes or other structures present barriers to diffusion. The kurtosis model
adds an extra term to the DTI model that is second order relative to the apparent
diffusion coefficient as shown in Equation (1.21).
Sk = S0 e−bk ADC + 16Kapp,k b2k (ADC)2 (1.21)
11
Where Kapp represents the kurtosis factor and determines the “peakedness” of the Gaus-
sian bell-shape [38]. If the kurtosis is zero, then diffusion is Gaussian and can be described
completely by the simple DTI model. However, if the kurtosis value is greater than zero,
then the diffusion profile has a higher peak, with smaller tails where diffusion is re-
stricted. In order to obtain this information, more b-values are required, as there are 21
independent parameters in the kurtosis model in comparison to only 6 in the DTI model,
implying longer scan time.
High Angular Resolution Diffusion Imaging (HARDI)
High Angular Resolution Diffusion Imaging (HARDI), as the name implies, requires a
very large number of unique diffusion directions – at least 60 directions to have a deter-
mined set (compared to six for DTI). In practice, HARDI scans, like most other diffusion
MRI, uses 120 or more directions at one b-value to generate an overdetermined set (com-
pared to typically 30 directions and one b-value in DTI) [39, 40]. HARDI attempts to
overcome the basic assumption of ellipsoidal Gaussian diffusion within DTI, at the cost of
a large number of diffusion directions. However, through the use of a high angular resolu-
tion on the diffusion direction sphere, HARDI is able to create a more flexible depiction
of diffusion over all directions, and is thus able to handle cases such as crossing fibres [7].
Many different methods have been proposed to visualize and identify the distribution of
fibres from HARDI data. A popular approach is to fit the diffusion distribution with
spherical harmonic functions, but other related model free approaches also exist such as
Q-Ball Imaging (QBI) [39, 41]. From HARDI datasets, it is common to compute the ori-
entation distribution function (ODF) – which describes the orientation of white matter
tracts and diffusion within the voxel.
Neurite Orientation Dispersion and Density Imaging (NODDI)
Neurite Orientation Dispersion and Density Imaging (NODDI) is another higher order
diffusion MRI model that incorporates more parameters than DTI, and in some respects
combines aspects of DKI and HARDI, requiring both multiple b-values and a large num-
ber of diffusion directions. Namely, NODDI requires a minimum of two HARDI “shells”,
12
where a “shell” is a set of diffusion directions at a specific b-value. The resulting image
intensities are modelled assuming three potential environments: intracellular (neurite)
space, extracellular space, and cerebro-spinal fluid (CSF) – with the underlying assump-
tion that each environment will inherently give a different diffusion MRI signal. The
signal behaviour from each environment is assumed to follow a particular model, and the
resulting image data is fit assuming a weighted summation of the models. The NODDI
model attempts to quantify the density of neurites within each voxel of the brain as well
as their orientation more accurately than DTI or HARDI are able to.
1.2.5 Clinical Uses of Diffusion MRI
Diffusion MRI is a tool that is integral to the diagnosis and staging of many diseases. Its
primary use clinically is for the diagnosis of stroke, but it can also be used to probe the
functional architecture of the brain – specifically white matter, as the highly myelinated
axons act as the “wiring” of the brain. Prime examples of diffusion MRI being useful
in the clinic would be cases of acute brain ischaemia [42], as the diffusion coefficient
decreases significantly within minutes of the ischaemic episode. Thus, one would see
a decrease in the MD as well in the FA of the brain region where the ischemia took
place in comparison to normal, while a standard T2-weighted scan is not diagnostic for
several hours [17, 43]. Similar effects are found in patients that have experienced strokes
[31, 44, 45, 46]. Diffusion-weighted MRI represents standard-of-care for diagnosis in these
cases.
Diffusion MRI has many additional clinical uses, such as measuring neuroanatomy
[47], stratification of brain tumours [48], prediction of cancer responses to chemoradia-
tion [49], and for visualizing effects of multiple sclerosis [50]. This is nowhere near an
exhaustive list, but shows the large range of pathologies that may be assessed based on
changes to water diffusion. The widespread use of diffusion MRI already promotes the
ease of implementation of new diffusion approaches, and motivates use of diffusion MRI
in research, both in human and small animal models.
It is common practice now within the clinic to utilize EPI as the imaging readout
for diffusion-weighted imaging as it is a very fast imaging technique. However, EPI is
13
very sensitive to susceptibility and spatial distortion artifacts, which motivate the use
of alternative readouts in some applications. Other strategies to speed imaging include
parallel receive strategies of various methods [51].
1.3 The Mouse as a Model
The mouse is a convenient mammalian model with high homology to humans both struc-
turally and genetically [52]. In addition, the mouse has an accelerated lifespan, and there
are a large number of genetically-modified (and characterized) mouse strains. While us-
ing the mouse as a genetic model is not within the scope of this thesis, it motivates
the technical developments in this research. Moreover, it has been shown that 90% of
mouse mutants with altered behavioural phenotypes have accompanying changes in neu-
roanatomy [53], making MRI an efficient and quantitative surrogate for evaluating the
mouse brain. Although the mouse also has a number of limitations – only a 10% fraction
of white matter volume compared to 50% in humans and a complete lack of cortical
folding [54] –it remains one of the most widely used models for neuroscience.
1.3.1 MRI in the Mouse
Due to the significantly smaller size of the mouse, there are some differences between
mouse and human MRI. In order to obtain similar “anatomical” resolution, voxel dimen-
sions must be scaled approximately 15-fold [55] to linear sizes of 100µm or smaller, which
translates to obtaining higher spatial-frequency information in k-space. An important
consequence to this is a loss in signal-to-noise ratio (SNR) [1], which is often recovered
by higher magnetic fields, custom radiofrequency coils, and longer scan times – for ex-
ample through acquisition of multiple averages. This issue is exacerbated for diffusion
imaging where a whole scan series must be acquired. One approach employed at the
Mouse Imaging Centre to increase efficiency is to image multiple samples simultaneously
so that, although each scan may be long, many images are acquired. This approach is
referred to as multiple-mouse MRI [56], and is especially attractive for imaging of ex vivo
brain samples, a strategy largely unique to preclinical research.
14
1.3.2 Diffusion MRI in the Mouse
Diffusion imaging for mouse data is often collected ex vivo due to the long scan times
required - it is not uncommon to have diffusion MRI scans taking 12 [57, 58] to 28
hours [59, 60]. Since the data is taken post mortem, there will be a general decrease
in the motion of water due to the loss of active transport of water within the brain
[61], but is still an excellent measure of relative diffusion within the sample. However,
even with the extended scan times and the post mortem acquisition, diffusion MRI in
the mouse provides an important contrast in ex vivo mouse research. It complements
laborious histological approaches – almost always restricted to select regions of interest
– by providing a whole-brain assessment to characterize white matter [62, 63] in models
of traumatic brain injury [64], stroke [65], or multiple sclerosis [66], as well as mapping
brain development and developmental pathologies through mouse models [67, 68, 69, 70].
It also, of course, provides a convenient link to a widely-used clinical tool, connecting
experimental work in the lab with observations available in patients.
The extended scan time of ex vivo mouse MRI, combined with the need for high-
resolution acquisitions of many different images, hampers the potential application of
methods such as DKI, HARDI, or NODDI in preclinical research. Finding a way to in-
crease the efficiency of these experiments would enable broad application for the scientific
community. This represents a classic trade-off in MRI, where spatial information must
be balanced against the number of images acquired to characterize another dimension
(such as contrast, time, frequency, or diffusion directions). With standard fully-sampled
image acquisitions, this trade-off is often unacceptable, requiring too low a resolution in
order to acquire the number of images needed in a given scan time. As an alternative,
I propose in this thesis an approach that improves the researcher’s ability to make this
trade-off by application of compressed sensing (CS), enabling a more flexible tuning of
the number of spatial samples and the number of diffusion directions.
15
1.4 Compressed Sensing (CS)
The amount of data that is required for higher order diffusion imaging methods is
significantly greater than the simple DTI case. In mice, where scan times are generally
increased already, the demand for scan time quickly becomes prohibitive. For example,
in high resolution DTI scans of ex vivo brains at our centre, 30-direction scans last for
12 hours. Running 120 directions thus requires a continuous 48 hours! This is sufficiently
impractical that it cannot be routinely performed.
In the various diffusion approaches described here, there are up to five dimensions
that need to be sampled: 3 spatial dimensions, 1 b-value dimension, and the diffusion
directions dimension (which in itself, of course, includes three directions). In standard
diffusion scans, the spatial dimensions are fully sampled, while the b-value and diffu-
sion dimensions are sampled at a very low resolution – often only one b-value and 30
directions over the Euclidean half-sphere. Such an acquisition is greatly undersampled
in the diffusion dimensions for higher order models. To remedy this, one approach would
be to instead undersample in the spatial dimensions – decreasing the amount of time
required for the acquisition of each specific b-value and direction combination – which
can then be leveraged to sample the required number of b-value and/or direction images.
While this would normally imply that the spatial dimensions will have lower resolution,
CS is an approach to image reconstruction that employs constraints to compensate for
undersampled data.
1.4.1 Introduction to Compressed Sensing
The theory of CS was first introduced by Donoho in 2006 as a method to reconstruct
complete data from sparsely-sampled sets [71]. Suppose there exists some unknown vector
x that can be measured in Rm. In a fully sampled case, at least m measurements would
be required to estimate x. However, for CS, if x can be assumed to largely consist of
zeros (i.e. is sparse) in some space, it is possible to make only n � m measurements,
and to obtain an accurate rendition of x by enforcing sparsity in the result.
The mode of undersampling is important in CS, so it is important that sampling
16
be planned to ensure certain conditions. Ideally, a method that acquires only from the
“important information” of the signal would be most efficient, but in practice a strategy
must be selected that works broadly for a given application. This requires establishing a
sampling space, the desired output space, and some sparse space (which can but need not
be identical to the output space). For MRI, all of the sampling occurs in k-space (also
denoted as “Fourier space”), and the desired output is in image space. To implement
the method, only a subset n of our m samples are collected in k-space such that n� m.
[71, 72, 73]. The CS algorithm has three major requirements [71, 72]:
a) The image is compressible (sparse) in some transform space;
b) Any artifacts due to the undersampling scheme are “noise-like” and do not
present as coherent aliasing artifact; and
c) The reconstruction method promotes sparsity in the selected space(s) and fi-
delity to acquired data.
The notion of having the data exist in a compressible space is readily met for the
MRI data that I am working with. Advances in compressing image data have been
occurring for years, specifically in the field of computing with respect to different ways of
compressing image files, such as the work done with JPEG-2000, which uses the wavelet
transform [74]. In wavelet space, most of the image can be described by a relatively
small number of high-intensity points, and the remaining points can largely be ignored.
There are a number of other transforms that define MRI data as sparse, and each of these
can be used to constrain the CS algorithm to exploit specific redundancies. A common
example is the Total Variation (TV). The TV is defined as an image of finite differences,
calculating a one-pixel gradient in image space [75]. The TV has significant intensities
only at tissue boundaries, and is close to zero elsewhere.
For MRI, the sampling in k-space is generally determined a priori, and for CS, must
be selected to ensure “noise-like” behaviour in image space. As mentioned earlier, n� m
samples will be acquired. Naively, one might undersample evenly across k-space; however,
in image space, this would cause an effective field of view (FOV) decrease, and cause
coherent aliasing due to sampling under the Shannon-Nyquist limit. The second naive
idea for undersampling, might be to take a low resolution image by only sampling the
17
Figure 1.2: A representation of the sparsity generated through the use of the total variation and wavelettransforms on a Shepp-Logan phantom. It is important to note that most of the values in each of thetwo sparsifying transforms are zeros. The total variation is excellent at determining the locations ofedges within the sample, and the wavelet coefficients are often used for compression of data.
centre region of k-space covered by n� m points. This would avoid the aliasing problem
of the first approach, but is likely to result in too poor a spatial resolution. In order
for CS to enhance the image, farther regions of k-space need to be sampled to obtain
information about the edges in the image. Instead, the current standard choice for
undersampling utilizes a (pseudo)-random sampling pattern in k-space. Namely, first a
small fully-sampled centre of k-space is acquired. On one hand, this ensures that there is
a good (low-resolution) basis to initialize the CS algorithm and on the other, it provides
information necessary to estimate the phase of the MR image (which is inherent in the
data acquisition but not handled as part of the CS reconstruction). Subsequently, the rest
of the sampled points (n − ncentre) are distributed through the remainder of k-space on
the same grid as the fully sampled k-space. This approach samples the high frequency
regions that give edge information to the image without generating coherent aliasing
artifacts and results in image-space artifacts that appear as noise.
The sampling scheme in this work will be constrained to a Cartesian grid, however
this need not be the case as many different sampling trajectories are used in the MR field
such as radial imaging [76, 77, 78], or spiral imaging [79, 80].
The simplest form of the CS reconstruction itself for MRI data is based on a con-
18
strained minimization described as follows [71, 72]:
minimize ||Ψx||1
s.t. ||Fux− y||2 < ε(1.22)
In this set of equations, x is our current guess for the reconstructed image (note that
later in Figure 2.7, our guess is in Wavelet space and is represented by ΨG); Ψ is a spar-
sifying transform, in this case the Wavelet Transform; Fu represents the “undersampled”
Fourier Transform (effectively applying a Fourier transform and only taking those points
acquired in undersampled k-space); y is the k-space data that was collected from the
scanner; and ε represents the tolerance for data consistency term. Finally, the ||x||n is
the `n norm of x, defined here as:
||x||p =
(n∑i=1
|xi|p) 1
p
for {p ∈ Z | p ≥ 1} (1.23)
||x||0 =n∑i=1
|xi|0 where 00 ≡ 0 (1.24)
Although the `0 “norm” in Equation (1.24) is in many ways the most intuitive mea-
sure of sparsity (i.e., counting the number of non-zero elements), it is computationally
impractical to work with. Instead, it is common to use the `1 norm, which has been
proven to be an acceptable solution to promote sparsity [81, 82, 83].
Equation (1.22) is a simple form of the CS equations and it is common to add addi-
tional terms to the minimization function, such as the TV term [72, 84].
minimize ||Ψx||1 + λ TV(x)
s.t. ||Fux− y||2 < ε(1.25)
The λ variable is used to tune the weight of the TV term relative to the wavelet sparsity
term. The equation is presented as such for the optimization:
argminx ||Fux− y||2 + λ1||Ψx||1 + λ2TV(x) (1.26)
19
Where now, λ1 and λ2 are used to tune the weights of the wavelet and TV terms relative to
the data consistency term, respectively. By utilizing other terms within the optimization,
just like the TV term has been added, this algorithm can be adapted for different types
of redundancies in specific data sets. Some examples could be redundancies in the time-
dimension, longitudinal anatomical data, or diffusion directions, which is the focus of
this thesis.
1.4.2 Current Uses of Compressed Sensing in Human Research
CS works best when SNR is high, and therefore it is more easily applied in human scans
than in mouse scans. As such, there is already extensive use of CS in human research.
For example, 3D contrast-enhanced angiography, in which the image is naturally sparse
and temporal frames evolve smoothly, is a perfect candidate [72, 85], and represents one
of the areas where CS was arguably first applied.
Although there is some interesting investigations of CS in combination with multi-
channel parallelization, CS is not used frequently for strictly anatomical work. CS is
optimal in cases where additional dimensions of redundancy exist, such as in dynamic
imaging, even on a slice-by-slice basis when imaging the brain [86]. Research has also
investigated the possibility to obtain higher resolution images through undersampled
datasets for functional MRI (fMRI) using spiral acquisition, where the redundancy from
the temporal dimension is exploited [87]. Using CS in conjunction with Sensitivity Encod-
ing (SENSE) for Chemical Exchange Saturation Transfer (CEST), allows for the overall
time required to be decreased by a factor of four (R = 4) [88]. The Human Connectome
Project has also been looking into using CS for diffusion MRI within the human brain in
order to obtain high resolution scans and an accurate mapping of the connections within
the brain [89], specifically with respect to Diffusion Spectrum Imaging (DSI), another
diffusion imaging approach requiring both a large number of b-values and directions. In
addition to this, there has been a recent interest in the use of dictionary-based sparsity
constraints exploiting previously acquired data to improve reconstruction even for factors
of R ≥ 4 [90].
The expansion of CS research within the human sphere has been rapid, now cover-
20
ing an immense number of potential applications. Specifically, in diffusion MRI, a CS
framework has been developed to estimate the degree of crossing fibres in the intra-voxel
environment within 30 direction DTI datasets [91]. In this experiment, CS is applied such
that mixing fractions are calculated based on the exploitation of the tensor directions
determined using DTI. This approach treats the diffusion direction dimensions as under-
sampled, and aims to discern information typically only available through a higher order
model like HARDI. Another experiment along these lines has attempted to use spherical
ridgelets to interpolate the data between diffusion directions to increase the resolution
of sampling in the diffusion directions dimension [92]. Each direction is fully sampled
and missing directions are inferred based on the expectation that each HARDI signal can
be expanded as a linear combination of spherical ridgelets. This model is effective, but
requires extremely high SNR to ensure the fits accurately reflect tissue properties, and
not image noise.
In diffusion MRI there are three fully sampled spatial dimensions, along with two
possible diffusion dimensions: diffusion directions and b-values, which are often sampled
at a much lower rate. However, in the current environment, where there is a significant
appetite for dense sampling in the diffusion direction or b-value dimensions, collecting a
large number of images with increasingly subtle intensity differences is required. An at-
tractive approach is to begin undersampling in the spatial dimensions in order to enable
increased sampling in the diffusion dimensions. For acquisitions based on Cartesian sam-
pling, it is convenient to undersample in only two dimensions because there is no benefit
gained from undersampling in the frequency encoding direction, which costs compara-
tively little time to acquire. Such an approach, in combination with a CS reconstruction,
could enable trade-offs in the amount of spatial versus diffusion information acquired, so
that more advanced diffusion models can be employed without significantly longer scan
time.
21
1.5 Aim
The purpose of this study is to exploit redundancy found in diffusion MRI data in con-
junction with CS to enable undersampling of k-space and a flexible trade-off of spatial
and diffusion data during image acquisition. This strategy was tested through several
stages. First, the CS algorithm was implemented and tested with anatomical mouse MRI
at a sampling rate of 25% to confirm function of the algorithm and to evaluate its per-
formance in isolated anatomical scans without an extra dimension of redundancy. Next,
significant modifications of a pulse sequence used for ex vivo diffusion MRI were made
to allow prospective acquisition of data for CS reconstruction. The modifications were
critical to addressing artifacts affecting the CS algorithm. Finally, the algorithm was
tested prospectively by acquiring a four-fold undersampled scan with 120 diffusion direc-
tions scanned (four times more than the baseline 30 directions) evaluating the benefit of
trading off spatial for diffusion direction sampling.
Chapter 2
Methods
2.1 Adaptation of Compressed Sensing for Diffusion
Data
The optimization scheme used was based originally off of a series of Matlab scripts
written by Dr. Michael Lustig to accompany his 2007 paper – however, the custom scripts
have been significantly adapted and ported into Python (https://python.org) [93] using
the packages NumPy [94], SciPy [95], and PyWavelets [96]. Matplotlib [97] was used for
visualization. The code was written so that the user can choose any of the several
available optimization routines and is parallelized over many cores for each index in the
readout direction (or other selected dimension). All optimization routines available for
the ported code are found in the SciPy package, with a conjugate gradient being the
default, and used for all CS reconstructions in this thesis.
The goal of my work is to collect high-angular resolution diffusion data in ex vivo
mouse brain samples to further elucidate diffusion microstructure within the brain for
mouse models of human disease. Using the baseline 30-direction diffusion scan at a
resolution of 78 µm, that runs in 12 hours in conjunction with four-fold undersampling
in the spatial dimensions, the high-angular information can be obtained through a 25%
spatially sampled set of 120 directions. However, in this implementation, the b0 scans will
remain fully-sampled due to their importance in subsequent computation of the diffusion
22
23
attenuation.
2.1.1 Total Variation in the Diffusion Dimension
The strategy I chose for enabling undersampling in the spatial dimensions was to enforce
similarity in nearby diffusion images as part of the CS constraint. There are multiple
approaches that could achieve this. However, as this strategy effectively seeks to mini-
mize variation in the diffusion direction dimension, it can be considered in many ways
analagous to the standard definition of TV in the spatial dimensions. Consequently, I
chose to expand the standard definition of TV to enable addition of the diffusion direc-
tion.
The design of a TV term for diffusion direction is complicated by the fact that each
diffusion direction is likely to have a large number of nearest neighbours. To account for
this, I assumed that over a small solid angle the intensity of the diffusion-weighted images
could be approximated as a linear function of vector direction. The gradient coefficients
of such an approximation represent a form of intensity gradient analagous to the spatial
TV terms. For any given direction, the directions considered “nearby” include those
that are nearly colinear, or those that are nearly 180◦ away. Using the image intensity
at each voxel and the difference in the diffusion vectors (i.e. creating a system where the
direction being analyzed is at the origin), a least squares fit can be used to determine
the intensity gradients in the diffusion direction space. Defined mathematically, for a
given direction r there are n closest directions [~dr,1, ~dr,2, ..., ~dr,n], where ~d represents the
diffusion direction for index n with respect to r. The local intensity is fit through a least
squares approach:
∆I =
Ir − Ir,1
...
Ir − Ir,n
(2.1)
24
A =
1, ~dr − ~d′r,1
...
1, ~dr − ~d′r,n
(2.2)
Aβ = ∆I (2.3)
β = (ATA)−1AT∆I (2.4)
= [β0 βx βy βz]T (2.5)
TVdiff =√β2x + β2
y + β2z (2.6)
~d′r,n is used to select ±~dr,n that generates the positive value for ~dr · ~dr,n. In
this representation ∆I is an n × 1 column vector, A is an n × 4 matrix to account for
a first order fit in the three spatial axes as well as a constant term, which leads to our
fit, β, a 4 × 1 matrix that represents the equation of the plane that fits the intensity
variation across diffusion directions at each voxel. It is expected that there should be
only nominal intensity offset (β0 = 0) since the selected directions are close and the
intensity distribution is assumed not to have any sharp discontinuities, however, it allows
an offset at the origin so that, for instance, the extremum of an ellipsoid is not penalized.
For the purposes of the objective function and the gradients, the defined TVdiff is
combined with the standard definition of TV. I added an additional tunable scaling
factor λdiff that can be changed to scale the weighting of TVdiff relative to the spatial
TV. For all of the work in this thesis, λdiff was set to unity.
TV = TVspatial + λdiffTVdiff (2.7)
2.1.2 Mathematical Approximations and Definitions
`1 Norm
The most intuitive constraint to enforce sparsity is to count the number of non-zero
entries. This is referred to as an `0 norm. However the `0 norm is difficult to use due to
25
its discrete nature, and difficulty in handling noise in data. Thus, it is convenient to use
the `1 norm [82]. As can be seen in Equation (1.23), at x = 0, the first derivative of the
`1 norm is discontinuous – following the curve of y = |x|. It is necessary to approximate
the `1 norm derivative because, often, the reconstruction is working in the realm where
x ≈ 0, at which point the derivative is ill-defined. In other applications of CS, the
approximation has been presented as [72]:
|x| ≈√x2 + µ (2.8)
d|x|dx≈ x√
x2 + µ(2.9)
Where µ is empirically chosen to limit the derivative near zero while providing a reason-
able approximation for abs(x). I chose an alternative approximation that easily tuned
and smooth through the origin:
|x| ≈ 1
aln [cosh(ax)] (2.10)
d|x|dx≈ tanh(ax) (2.11)
Where a is a user-tuned variable to effect the sharpness of the curve, as can be seen in
Figure 2.1. When a larger value of a is used, the curve is nearly identical to |x|. This
method allows for some tunability in the effects of very small values (where the curve is
smooth), and retains the benefit that it is continuous everywhere. For all calculations, a
value of a = 25 was used.
TV Kernel
The spatial TV terms were implemented as a convolution of a gradient kernel and the
image. This allows flexibility in how this term is applied in the algorithm, by allowing
for different weighting to surrounding pixels based on the kernel. This method can be
readily implemented for the standard TV method through the following kernels:
26
Figure 2.1: (a) Shows a comparison between different mappings of 1a log(cosh(ax)) for increasing values
of a when compared to |x|. Note that for larger values of a the curve approximates |x|. (b) Shows acomparison between the derivatives of the above functions, represented by tanh(ax) and sgn(x).
TVx =
0 0 0
0 −1 1
0 0 0
TVy =
0 0 0
0 −1 0
0 1 0
(2.12)
In future implementations, it may be useful to apply a sparsity term with the third
dimension as well [75], which would require equivalent definition of TVz, or exploration of
alternative gradient kernels for edge detection – such as the Gabor filter – in the sparsity
constraint [98].
2.1.3 Phase in CS Reconstructions
Note that the CS reconstruction is defined using a real image, whereas MRI data is
complex with both magnitude and phase. One must therefore include a step in the CS
algorithm to account for phase prior to computations of sparsity in image space, but
reapply it for comparison with data collected in k-space. Indeed, the most important
initialization step for CS reconstruction is arguably the phase estimation. When accurate
27
phase is used, reconstructions can be accurate with only 8.3% of the data in an idealized
phantom [72]. However, to have a perfect phase estimation, one requires a complete
image, which of course is contradictory to the notion of undersampling in the first place.
The primary method to obtain a phase estimation is thus to use a blurred, low-resolution
image attained by fully sampling a small region at the centre of k-space. While this
provides a good phase estimate in many cases, edge information is inevitably lost and
cannot be recovered in the reconstruction.
In the work presented here, I elected to map the phase based in large part on the fully
sampled b0 images. This approach, in principle, gives the highest resolution phase map
available. However, significant efforts were required to enable this strategy, owing to the
sensitivity of RARE to anomalous phase and the tendency of the diffusion gradients to
introduce anomalous phase. This is elaborated on in Section 2.3.3. One of the primary
areas for future research should be optimization of the phase estimation of undersampled
data, which would broaden the scope of CS applications.
2.2 Sampling
For all of the sampling schemes discussed within this paper, a cylindrical acquisition
scheme was used [55]. In this approach the corners outside a fixed radius in the plane
defined by the two phase-encode directions are not collected – thus only π/4 of the
full Cartesian grid is sampled – or a reduction of ∼22%. All references to percentage
sampling within this work are in reference to this baseline (i.e., 25% sampling indicates
0.25 × π/4 sampled points of the fully-sampled Cartesian grid). Only the two phase-
encode directions will be undersampled, as there is no benefit to undersampling in the
frequency-encode direction.
Generally, the sampling scheme in undersampled regions of k-space can be uniform or
variable density. If uniform undersampling is used, a probability density function (PDF)
is applied such that the probability of a point in k-space being sampled is equal for all
points (with the exception of the fully sampled centre). This sampling method represents
low- and high-frequency image information equally, however, it is also prone to artifacts
28
from omission of low-frequency data and from a discontinuous PDF [72]. Variable density
undersampling on the other hand, focuses in on the fact that most of the signal can be
found closer to the centre of k-space, and attempts to acquire more of it at the expense
of information closer to the edge. As such, it applies a PDF that drops proportionally to
r−p from the centre of k-space, where p ≥ 1, generally bounded between 1 and 6. This
method of sampling has been shown to converge faster in the CS algorithm for cases
where no extra redundancy is present, such as in anatomical scans [72].
For the purposes of this thesis, all sampling schemes used will be at 25% sampling of a
fully sampled cylindrical acquisition. In addition, an alternative method of sampling will
be presented that was selected a priori in order to exploit redundancy across multiple
diffusion directions in the sampling in order to optimally sample k-space across diffusion
directions.
2.2.1 Choosing Directions for Diffusion Acquisition
Diffusion directions were selected based on a simple algorithm that treated each direction
that I wanted to probe as a pair of point charges constrained to the unit sphere – that
is to say, in spherical coordinates, points were placed at (ρ = 1, φ, θ) and (ρ = −1, φ, θ).
The use of two “charges” is motivated by the notion that in diffusion MRI, a positive
and negative diffusion gradient probe equivalent directions.
The points were initially placed on the sphere following a method that chose latitudes
at regular intervals, and then placed a varying number of points based on the size of the
cross sectional circle at that point of the sphere [99]. Using this starting point, the points
were moved on the sphere in order to minimize the following energy:
E =n−1∑i=1
n∑j=i+1
max(| ~ri ± ~rj |−1) (2.13)
A conjugate gradient system was used to solve this process with numerical calculation
of the gradient. Using this method, a set of equispaced directions on a unit sphere are
generated that can be used for uniform coverage of the unit sphere.
29
2.2.2 Spatial Undersampling Across Diffusion Directions
It is convenient to have a good initial guess for the CS acquisition based on acquired
data. Since diffusion-induced intensity loss varies slowly with diffusion direction, k-space
from nearby directions can be used in place of unsampled data to generate an estimate
of the fully-sampled image. In order to utilize information between different diffusion
directions, I designed an undersampling scheme spread uniformly across directions, such
that a collection of nearest neighbour directions together had a nearly fully-sampled
image. To satisfy this requirement, a hybrid of variable density undersampling and
uniform density undersampling was used.
The sampling selection is performed once for an individual protocol with each diffusion
direction assigned its own unique sampling. The centre of k-space was fully sampled, and
then uniform density sampling was used around the periphery of k-space. In order to
ensure that there were no structured discontinuities in k-space at the border between
these regions, a “taper” region was included at the boundary. This can be seen in Figure
2.4(b) where three major regions exists:
1. Fully sampled centre
2. Variable density (“taper”) region
3. Spatially uniform undersampling constrained in distribution across the diffusion
directions
In order for neighbouring directions to have, when combined, a uniform sampling of
k-space, the sampling of neighbouring directions should be non-overlapping. Hence, a
second optimization method using simple electrostatics was implemented to minimize the
potential energy of groups of charges placed on the unit sphere at locations described by
the diffusion directions. One assumption in diffusion MRI is that diffusive motion runs
along both the positive and negative directions on a given axis, so that equal and opposite
diffusion directions are equivalent. For optimization, I therefore considered these points
equivalent. Consequently, the objective function was defined as follows:
30
Figure 2.2: Probability density functions for CS undersampling: (a) Variable density undersampling, asis commonly used in CS reconstructions, shown both in 2D and a 1D line that is radially symmetric overthe sampling pattern. This method only has the fully sampled centre and the variable density region (b)The novel hybrid pattern with a central region which has both the fully sampled region and the variabledensity region exactly as before, as well as the external region where directionally biased sampling isused which is equivalent to uniform density sampling.
31
obj =
q∑g=1
m−1∑i
m∑j=i+1
max(| ± ~rgi − ~rgj|−2) (2.14)
In this equation, g represents a group from the set of q total groups, with a total of
m members per group. The max() function accounts for the symmetry of diffusion along
the reference axis. For even a modest number of diffusion directions, this function has a
large number of local minima. A brute force method could be employed to find the global
minimum, however the number of different groups quickly becomes intractable because of
the binomial nature of combinations. Moreover, it is not essential that a global minimum
be identified for our purpose. As such, an alternative method was created and dubbed
“Team Picking”. The “Team Picking” algorithm is similar to children choosing teams
on the playground where the most favourable of the as yet unselected choices is made.
In this algorithm, directions are selected to fill q groups, where q is determined by
the number of directions (n) and the percentage of k-space to be sampled outside the
variable density sampled region, (p). Each group in k will have m or m+ 1 directions.
q = bp−1c (2.15)
m = bnpc (2.16)
Where b·c represents the floor function. The algorithm proceeded as follows: first start
with choosing a single direction – this becomes the first direction in group one. Then
the q − 1 directions closest to that direction are found. These will be the “captains”,
populating the first direction in groups 2 through q. With the remaining directions,
unique directions are added to each group in order to minimize the objective function
with each addition. Once one minimizing direction is found per group, they are added
to the corresponding “team” and used to calculate the objective function for all future
iterations. This process is then repeated until the teams are populated. By using this
algorithm, it is ensured that directions close to one another generally do not collect the
same data – and that over some small solid angle there exists a fully sampled k-space,
as can be seen in Figure 2.3.
32
Figure 2.3: (a) Different “teams” of directions created using the team building method with p = 0.2and n = 30. Each direction is shown as a vector going from one end of the sphere to the other throughthe origin. Note that in this case, as this is only one example beginning with one direction, some ofthe members on a specific team (i.e. blue) are close together. (b) Representation of how k-space issampled, where white is the fully sampled region, grey represents variable density sampling and thecolours represent the data that will be collected by each team based on colour. Over a small solid angle,the entire region of k-space is sampled.
2.3 Implementation of ex vivo Diffusion RARE
2.3.1 Acquisition Parameters
To test the initial implementation of the CS algorithm independent of the added diffusion
component, high-resolution anatomical images were retrospectively undersampled and
reconstructed via the CS algorithm. This data was acquired using a 7.0 T Bruker MRI
(Bruker Corporation, Billerica, USA) with a 30 cm bore. A cylindrical k-space acquisition
[55] in a standard gradient-echo sequence was used with the following parameters: TR
26 ms, TE 8 ms, flip angle of 37◦, matrix size 334 × 294 × 294, with 2 averages, and 1
hour acquisition time. Twenty-four hours prior to imaging, the mice were injected with
50mg/kg MnCl2 to enhance contrast for T1-weighted imaging.
After confirming the algorithm appeared to be working, I established a sequence and
protocol for imaging ex vivo brain specimens. For this work, a multi-channel, 7.0 T MRI
magnet with a 40 cm diameter bore (Magnex Scientific, Oxford, UK) with an Agilent
Direct Drive console (Varian Inc., Paolo Alto, CA, USA) and Resonance Research Inc
(Billerica, MA) gradient system was used. Samples were placed in a custom 16-coil (4×4)
33
solenoid array. In this configuration, the corner coils are not used for diffusion imaging
due to significant differences in gradient strength at the edge of the field [56, 100].
A diffusion imaging sequence was implemented for prospective acquisition of under-
sampled data. The diffusion sequence was based on a 3-dimensional RARE sequence,
which improves efficiency by collecting several k-space lines after a single excitation, us-
ing a train of refocusing pulses to repeatedly recover otherwise dephasing signal. For our
diffusion acquisitions, a 6 echo train was used with the centre of k-space acquired on the
first echo. A schematic of the pulse sequence used can be seen in Figure 1.2. The scan
parameters were TR 300 ms, TE 15 ms, TEeff 30 ms, 90 µm isotropic voxels, FOV of
1.62 × 1.62 × 2.91 cm, and a matrix size of 180 × 180 × 324. In the fully sampled 30
direction case, this scan takes approximately 12 hours to acquire, which implies that a
120 direction acquisition would take 48 hours. With the use of undersampling and CS,
the undersampled 120 direction case also takes approximately 12 hours to acquire.
As described, RARE provides improved efficiency by repeated refocusing pulses for
recovery of signal and acquisition of additional k-space lines. This has a few implications
that became integral to this work. First, each phase encode grid point in k-space must be
assigned to a particular repetition time and echo in the acquisition. Second, the phase of
the signal down the echo train must be carefully controlled and consistent. Echo-to-echo
inconsistencies in phase – either globally or linearly varying across the sample – lead to
prominent artifacts. Owing to the refocusing pulses in a RARE sequence, the erroneous
phase often separates between even and odd echoes. These are discussed further in the
next two sections.
2.3.2 Phase Encode Partitioning Scheme
Having selected a k-space acquisition scheme that omits the corners outside a given
radius, the remaining grid points in the PE1-PE2 plane must be divided across the (six)
echos of the echo train. The circular geometry lends itself to a partitioning based on
radius, with each echo separated into an annulus as can be seen in Figure 2.4(a). The
undersampling pattern described above must be combined in the context of this RARE
acquisition, resulting in a modified version of the PE1-PE2 sampling as a function of
34
echo as shown in Figure 2.4(b).
Maintaining strict control of phase down the echo train is critical in the RARE pulse
sequence. One source of error is residual phase from phase-encode gradient pulses. These
may take the form of either a constant phase over the whole sample, or residual phase
gradients across the sample. Given good quality refocusing pulses, the former can be
corrected simply by measurement of the residual phase and correction in post-processing
prior to reconstruction. This is conducted as a standard part of our ex vivo RARE
acquisition [55], and was employed here as well. Residual phase gradient terms after
standard imaging pulses are sufficiently small that this effect can be neglected.
Unfortunately, the addition of diffusion gradients complicates things considerably.
The diffusion gradients are much larger than the imaging gradients, and can occur in any
orientation (determined by diffusion directions). Consequently, the potential for residual
phase and phase gradients is greatly increased. Indeed, our first implementation showed
significant artifact resulting from echo-to-echo inconsistencies in k-space acquisition. In
principle, induced phases that are uniform can be handled with the same retrospective
approach described for the phases induced by imaging gradients. Conversely, residual
gradient terms after the diffusion gradient are much more problematic because they have
the effect of shifting the effective position in k-space of the acquired data, so that it no
longer lies on a simple Cartesian grid. The most obvious way to handle this problem
might be to empirically tune gradient waveforms to nullify any residual gradient terms.
However, I found that this approach was not feasible over the whole gradient bore,
because some positions required different tuning than others. Thus, I instead designed
an approach to accommodate this problem as detailed in the next section.
2.3.3 Correction of Diffusion-Gradient Induced Phase
Interleaving ±~d Acquisitions: The “Checkerboard”
Imperfect behaviour of the magnetic field gradients is often attributed to eddy currents,
but can also be related to gradient hysteresis, concomitant fields or other unanticipated
behaviours. The resultant phase or phase gradients affect the subsequent acquisition,
35
Figure 2.4: (a) Radially segmented splitting of echos shown in the fully sampled case to show how theannuli are split (b) A representative undersampled table using radially segmented k-space while stillcollecting the centre of k-space on the first echo. (c) Fully sampled example of how a radially segmentedphase encode table would be collected if the checkerboard pattern was used. The zoomed in region showshow each point alternates as to whether it is collected on ~d or -~d for each echo. (d) The undersampledversion. Note here that though the points are not all collected, the same checkerboard pattern is followed,where ~d and -~d are collected on the same mapping as in the previous figure.
36
and is particularly problematic with a RARE readout, and became a major hurdle for
my implementation with ex vivo diffusion imaging.
In order to accommodate the problem of diffusion-gradient induced phase, I employed
both positive and negative diffusion gradients for each direction, in a pattern that alter-
nated in a “checkerboard” across k-space (see Figure 2.4(c) and (d)). Ideally, diffusion
in each direction should be identical, so this pattern should make no difference to the
resulting image. If, however, there are induced phase or gradient errors, these produce
a similar “checkerboard” pattern in k-space, which manifest in image space as ghosts in
the corners. This has two potential benefits. First, comparison of separate reconstruc-
tion of the “dark” and “light” checkerboards with the b0 reference allows a first-order
estimate and correction of these terms. Second, residual phase errors are more likely
to produce corner ghosts instead of characteristic ripple at edges in the image, a more
palatable ghosting pattern. In this way, the checkerboard sampling pattern allows for the
acquisition of data to correct for first-order problems induced by the diffusion gradients.
Diffusion-Gradient Induced Phase Corrections
Estimation and correction of diffusion-gradient induced phase was achieved by compar-
ison to the fully-sampled b0 reference scans. Correction of phase is applied in several
steps.
First, RARE phase corrections as described above are applied to both the b0 and b
data. The b0 data, which included 5 images acquired uniformly through the acquisition,
was averaged to create a high SNR reference, called b0. From b0, a mask is created
encompassing the brain tissue (by a simple threshold operation). This mask defines the
region of interest over which phase match will be evaluated.
Next, the TRs are split into the light and dark checkerboards (which correspond to
the positive and negative amplitude diffusion gradients) for both the b0 and the b data.
Since the b0 data is fully sampled, when it is split into the checkerboard pattern, the
image has perfect ghosts in the corner that can be joined to generate an exact replicate
of the brain in the centre. Within the defined mask, the phase of the b images is compared
to the respective b0 reference. The desired phase fit Φ is split into four parts:
37
1. Constant Phase (φ0)
2. Frequency Encoding (Readout) Phase Gradient Term (φ′RO)
3. PE1 Phase Gradient Term (φ′PE1)
4. PE2 Phase Gradient Term (φ′PE2)
In this notation, φ′ is used to indicate a gradient. The constant phase is fit by
measuring the difference in overall phase for the regions of the images that fall within
the centre mask:
φ0 = ∠(∑
ej(∠Ib=0−∠Ib6=0))
(2.17)
The three phase ramps are all calculated in a similar manner to each other, where
the two datasets are projected onto the axis in question by summing across the other
two dimensions. The data is then zero padded to and Fourier transformed to determine
the distance between peak amplitudes, which maps directly to a phase ramp term. This
approach provided satisfactory results and was faster than other alternatives tested.
Subsequently, the derived coefficients are used either as a preprocessing correction (for
the constant and readout terms) or during the CS reconstruction (for the phase encode
gradient terms). In both cases, the corrections take into account the applied diffusion
gradient direction (i.e., light vs. dark checkerboard) and even vs odd echo.
The latter is necessary because each subsequent refocusing pulse reverses the phase,
and the applied correction must follow suit. Thus, each point in k-space is corrected
based on its (L)ight/(D)ark (i.e. ~d/-~d) and (E)ven/(O)dd echo assignment as follows:
1. Φ(L,O) = ΦL
2. Φ(L,E) = −ΦL
3. Φ(D,O) = ΦD
4. Φ(D,E) = −ΦD
Where Φx = [φ0 φ′RO φ′PE1 φ′PE2]TL/D. φ0 and φRO can be applied prior to CS because
the constant phase is applied element-by-element and the RO direction is fully-sampled.
38
The two phase encode directions are undersampled, however, and the phase gradient
terms need to be included as part of the CS algorithm. It is therefore convenient to
define Φx = [φ′PE1 φ′PE2]TL/D, which will be used in Figures 2.5 and 2.7.
2.4 CS Reconstruction with ex vivo Diffusion RARE
The algorithm that the CS reconstruction follows is shown schematically in Figure 2.7.
The first step is to apply a FFT down kRO which is fully sampled. For convenient parallel
processing, the reconstruction is then performed on a slice-by-slice basis (taking “slice”
to indicate an index in the read direction), reducing the overall time required to complete
the reconstruction, and spreading the memory requirements. The latter of these points
becomes important as the number of directions increases.
2.4.1 Slice-to-Slice Correction
Introducing the slice-by-slice method of reconstruction is convenient for parallelization,
but can introduce slice-to-slice variability in overall intensity. To correct this, the final
volume of each direction undergoes intensity scaling in the frequency-encode direction
based on the intensity profile of the undersampled data from the scanner (i.e. FFT
reconstruction without the use of CS), as is shown in Figure 2.8.
2.5 Mice
Wild type CD1 mice at age P14 from the Centre for Phenogenomics (TCP) in-house
colonies were used for the in vivo anatomical CS experiments. For the diffusion biased
CS experiments, wild type C57/Blk6 mice were perfusion-fixed at age P14. These mice
were from a study of the Syngap1 gene (as provided by Dr. Gavin Rumbaugh). For all
animal experiments, approval was obtained from the Centre for Phenogenomics Animal
Care Committee.
39
Figure 2.5: A flowchart representing the ±~d phase corrections required to correct diffusion-gradientinduced phase. This figures expands 1© in Figure 2.6. In this method, the constant phase and thegradient along the read direction (RO) are applied to generate modified k-space data (kaq), while thephase-encode phase gradient terms are fed to the CS algorithm.
40
Figure 2.6: A simplified flowchart of how to acquire and correct the data for CS reconstruction. 1© canbe found in Figure 2.5, and 2© can be found in Figure 2.7.
41
Figure 2.7: A simplified flowchart demonstrating the CS algorithm. This figures expands 2© in Figure 2.6.The diffusion specific addition for this project is captured within the left arm of the Total Variation. ΨG
represents the current guess (represented in wavelet space) that is being tested through the algorithm.
For each of the boxes, both the objective function f and its gradient ~∇f are calculated. For theData Consistency box, the ‘mask’ is the sampling mask used in acquisition, and determines which phaseΦ(B/W,E/O) is to be applied. The wavelet box is simple because the current image guess is representedin wavelet space. Note here that when ΨG is updated, a scalar c is applied to the gradient that representsthe line search term.
42
Figure 2.8: Slice-to-slice intensity corrections. In this image, the readout direction is in the left-to-rightdirection. (a) One representative slice to show post-CS slice-to-slice differences. (b) Intensity profiledifferences used to scale each slice. Red is the intensity profile (summed over the two phase encodedirections) of the post-CS reconstruction and blue represents the pre-CS intensity profile. (c) The sameslice after intensity correction as per the profile correction.
Chapter 3
Results
3.1 CS Reconstruction – Proof of Principle
3.1.1 Retrospectively Undersampled Shepp-Logan Phantom
As a proof of principle, the CS implementation was tested using a digital phantom and
simulation of an undersampled (anatomical) acquisition. The digital phantom was gen-
erated from the Shepp-Logan phantom in Matlab (The MathWorks, Natick, USA) with
the addition of smoothly varying phase and complex noise. The data was retrospectively
undersampled from the complex dataset in k-space and run through the CS algorithm
(using the phase from the fully sampled case), as in [72]. The CS reconstruction reduced
the incoherent noise present from the zero-filled dataset, reconstructing even the three
small light-grey regions close to the bottom of the phantom as shown in the blue box in
Figure 3.1.
3.1.2 Retrospectively Undersampled Anatomical in vivo Data
A retrospective undersampling and CS reconstruction was used to assess the algorithm’s
performance in mouse data using in vivo data collected from the Bruker MRI. The data
was retrospectively undersampled using the variable density undersampling strategy at
25% sampling. If the CS-based acquisition had been acquired prospectively, it would
have taken approximately 7.5 minutes (as compared to 30 minutes for the fully-sampled
43
44
Figure 3.1: Example of a Shepp-Logan phantom with 10% complex Gaussian noise added run throughthe CS algorithm at 25% sampling (R = 4). Parameters used: λ1 = λ2 = 0.05.
dataset). For the CS reconstruction shown below, one average of the fully-sampled data
was used to generate the phase map for the CS reconstruction. Without the use of this
phase, the reconstruction does not perform nearly as well. In this particular image, this
can be largely attributed to a large phase inhomogeneity at the upper left side of the
image, which appears darkened in the zero-filled data, and even shows minor artifact in
the fully-sampled image.
In the reconstruction, we can see that the post-CS data appears qualitatively better
than the zero-filled case owing to the removal of most of the incoherent sampling noise.
While the CS reconstruction on anatomical data alone does not recapitulate the anatom-
ical image, it provides an excellent quality with much of the detail in a fraction of the
time.
3.2 Diffusion MRI Data
After demonstrating that the algorithm was functioning in anatomical experiments, I
wanted to determine if CS could trade-off spatial sampling for an increased number of
diffusion directions. The scan that was used as the starting (and comparison) point was
is a 30 direction diffusion MRI scan with a b-value of 2000 s/mm2 and five b0 scans. The
total acquisition time is 12 hours for the fully sampled cylindrical acquisition of k-space.
45
Figure 3.2: A test of CS reconstruction in an in vivo anatomical image of the mouse brain. At left, afully-sampled in vivo brain image is shown (inset at bottom). Undersampling (25% of the total data)results in the image in the middle column, where missing data is replaced with zeros. At right, the CSreconstruction is used to recover detail. Not the partial recovery of the subdural fat (blue arrow, toprow) and a blood vessel in the cortex (blue arrow, bottom row) through CS. Parameters used: a = 25,λ1 = λ2 = 0.05.
3.2.1 “Checkerboard” Correction of Diffusion-Gradient Induced
Phase
The diffusion gradients induce substantial residual phase. To test the “checkerboard”
approach – which uses interleaved acquisitions of positive and negative diffusion gradients
– for measuring and eliminating this phase, I first examined fully-sampled diffusion-
weighted images acquired in this fashion. An example is provided in Figure 3.3(a).
In this image, the phase inconsistencies between the ~di and −~di directions manifest as
ghosts in the corners of the image. In order to correct the phase differences between
the ~di and −~di acquisitions in the checkerboard data, the phase of each is mapped to
the corresponding b0 data (splitting of the b0 data by the same pattern). Application
of the obtained phase corrections then enables generation of a clean image where the
corner ghosts are minimized. Successful application of this method provides an approach
for minimizing the impact of diffusion-gradient induced phase errors and, where present,
manifests them as corner ghosts as opposed to edge ripple artifacts.
46
Figure 3.3: Diffusion specific phase corrections from ±~d checkerboard data. (a) A diffusion (b > 0)dataset after RARE corrections applied. RARE corrections are not sufficient to remove the ghostingin the corner caused by the ±~d differences. (b) The top and bottom row are the ±~d data split fromeach other, respectively. The column on the left is the b data, and the column on the right is the b0data. The phase from the b data is fit to the respective b0 phase on the right. (c) After phase correctionas described in Section 2.3.3 the corner ghosts are substantially reduced, and the phase of the b datamatches the b0 phase.
47
3.2.2 Compressed Sensing Reconstruction
With the CS algorithm demonstrated and a new approach for phase correction imple-
mented, I prospectively acquired a 120-direction diffusion-weighted data set with four-fold
undersampling. The sampling scheme is described in Section 2.2.2. In the same acqui-
sition time as the fully-sampled standard, the CS image acquisition and reconstruction
is able to obtain 120 diffusion directions acquired at the same b value in addition to the
five fully-sampled b0 scans. Thus by implementing this method, there is four times the
amount of diffusion direction data to work with, within the same acquisition time.
Figures 3.4 and 3.5 are representative examples providing comparison of individual
directions between fully-sampled images and the undersampled image acquisitions. The
latter are show both pre- and post-CS-reconstruction. For each CS reconstruction, the
tuning factors λ1 and λ2 were set to a value of 0.05 relative to the data consistency term.
Post CS-reconstruction, it can be seen that on an individual diffusion direction level,
at 25% sampling, there is some reduction in image quality compared to a fully-sampled
acquisition. For the duration of this work, I will be using the correlation as a measure of
image quality when comparing the fully-sampled, zero-filled, and post-CS datasets. As
a point of reference, the correlation between b0 data was 98.6% ± 0.1% across the five
b0 scans, where the error represents the range of data about the mean. The correlations
were measured over a mask covering only the brain in each image.
When applying undersampling and CS, the reduced quality is perhaps most evident
in the cerebellum, where fine details of foliation remain somewhat blurred in the CS
reconstruction. However, the CS-reconstruction provides significant improvement over
a zero-filled reconstruction. This may be most obvious in the corpus collosum. The
correlation across volumes for the pre- and post-CS data were 82.1% ± 5.1% and 87.4%
± 3.9% respectively.
The comparison in Figure 3.4 is biased, in the sense that the fully-sampled image in (b)
took four times longer to acquire than the CS reconstruction in (d). A comparison based
on four nearby directions from the undersampled acquisition to the fully-sampled result
was therefore also performed. This comparison is provided in Figure 3.5, which shows
48
Figure 3.4: Comparison of fully-sampled and undersampled image reconstructions for an individualdiffusion direction. (a) A schematic display an outline of the brain in three orthogonal orientations withslices indicated the location of planes in (b) through (d). Double-headed arrows indicate the diffusiondirection probed in that plane. (b) A fully-sampled image (from a 30-direction DTI acquisition) shows
the baseline image result (note that this is without the use of the ±~d checkerboard acquisition scheme).
(c) Undersampled data with ±~d checkerboard acquisition scheme before undergoing CS reconstructionand in (d) after CS reconstruction. cer. represents the cerebellum and cc represents the corpus callosum
49
the four individual undersampled images (a-d) and their average (e). When evaluated
in this way, the undersampled CS reconstruction performs very similarly to the fully-
sampled acquisition. Although some loss of detail remains (again, see the cerebellum),
the quality is very comparable, yielding a correlation of 94.2% ± 2.5%. The benefit of
the undersampling data begins to be evident in this figure, where the additional data
from multiple undersampled images (a-d) provides a basis for signal intensity analyses
that cannot be performed with one image. This increased diffusion direction resolution
is the basis for evaluating more subtle differences in intensity as a function of diffusion
direction.
In order to compare the 120-direction images with the 30-direction data for DTI
analysis, I computed FA maps across the entire brain for both cases. In Figure 3.6,
the FA maps are shown. Here it can be seen that the FA between the two datasets
exhibits anatomically very similar features. Interestingly, there is a general decrease in
the FA in grey matter in the 120-direction case, with a reduction from values around
0.4 using the original method, to 0.25 in the new one. In fact, the reduced value is
more consistent with expectations for grey matter. The decrease in FA is further shown
in a histogram comparing the FA between the fully sampled data and the CS data in
Figure 3.6(c). I suspect that this drop is largely due to the phase corrections enabled by
the checkerboard acquisition, which corrects diffusion-gradient induced phase errors and
provides a substantial reduction in artifact levels in specific directions.
3.3 Potential Use for HARDI Analysis
The real benefit of the CS undersampled acquisitions is the much increased diffusion
direction information. This is exemplified in Figure 3.7, where an individual slice from
all directions is shown, adjacent to a schematic showing all directions sampled for both the
fully-sampled (a) and undersampled (b) image series. The substantial increase in number
of directions enables more complicated mapping of the directional dependence of diffusion
than with DTI. As a proof-of-principle, a simple HARDI analysis using a Q-ball method
has been performed using the same data presented in Figure 3.7. The initial results of
50
Figure 3.5: Comparison of fully-sampled and undersampled image acquisitions with matched acquisitiontimes. For each of these images, the arrows on top of each column represent the direction of the diffusiongradient in that plane. (a)-(d) Representative CS-reconstructed images acquired with 25% sampling areshown for the four closest directions to the fully-sampled direction in (f). (e) The average of images(a)-(d) provides a representative comparison to the fully sampled case. cer. represents the cerebellumand cc represents the corpus callosum
51
Figure 3.6: Comparison of FA maps in (a) 30-direction fully-sampled and (b) 120-direction undersampledimage acquisitions are shown. Anatomical features are well-preserved between the two images, and mayeven be enhanced in some features of (b). An overall decrease in the FA of gray matter in the 120-direction case is attributed to improved handling of artifacts by the “checkerboard” phase correctionscheme and by the larger number of directions involved in the fit. (c) Comparison of FA values in 30-direction fully-sampled (FS) and 120-direction CS acquisitions are shown. A stark decrease can be seenin the FA values for the 120-direction case, which is likely due to spurious acquisition in the 30-directiondata which is corrected through both the higher angular resolution and the checkerboard acquisitionscheme.
52
Figure 3.7: In the left column, the diffusion directions are plotted as vectors going through the originand touching the unit sphere on either side, representing the resolution in the dimension of diffusiondirections. In the right column, a representative slice is taken for each direction to visualize the sizes ofthe two datasets between (a) the original 30-direction method and (b) the novel method that incorporatesspatial undersampling and CS.
53
this analysis are shown in Figure 3.8, which highlights the corpus callosum (a) and the
anterior commisure (b). With the anterior commissure and corpus callosum the overlaid
representations of the diffusion exhibit a much more elongated profile, produced primarily
by reduction in the perpendicular directions. Interestingly, many voxels (particularly at
boundaries of white matter) exhibit patterns that cannot be well described by a DTI
model. The underlying cause of these changes would interesting to study further. It has
been shown previously that approximately 90% of white matter has some level of crossing
fibers [101], and and this approach provides an opportunity to investigate such features.
While I have focused on a HARDI analysis at first, implementation for other models of
diffusion is possible as well.
54
Figure 3.8: HARDI analysis of 120-direction undersampled diffusion MRI data. Anatomical imagesshowing the relative location of the enlarged ROIs are presented in the left column, with the rightcolumn showing zoomed regions with an HARDI analysis through an overlay of the ODF as calculatedby QBI (using DiPy [102]) represented as a deformed sphere based on the spherical harmonics on eachvoxel. Likelihood of diffusion in a direction is presented both by sphere deformation and colour, wherered implies strong diffusivity and blue represent low diffusivity. The specific regions that are probed are(a) corpus collosum (cc) and the anterior commissure (ac), and (b) anterior commissure (ac). Such ananalysis cannot be performed using the 30-direction fully-sampled acquisition.
Chapter 4
Discussion
In this thesis, a method was developed to permit a trade-off of spatial and diffusion
direction resolution in diffusion-based MRI analyses. The approach was further designed
to be acquired using a self-calibrating sequence by accounting for gradient imperfections
that induce spurious phase. The image data was then reconstructed using a model
that emphasized data similarity between diffusion directions close together on the unit
sphere. This enabled a large number of directions to be acquired, providing the potential
for accelerated acquisition of data sets appropriate for HARDI analysis. In this chapter,
I discuss the features and limitations of the proposed approach.
4.1 Phase Corrections
RARE acquisitions are notoriously artifact prone due to their extreme sensitivity to phase
errors through the repeated use of refocusing pulses down the echo train. As a result, the
use of this pulse sequence adds a layer of difficulty for the CS implementation, especially
since a high-quality phase estimation is critical for accurate CS reconstruction. I handled
this phase problem with a “checkerboard” acquisition strategy, in which the symmetry of
diffusion and properties of the Fourier transform were exploited to move artifact to the
corners of the image. Comparison to the b0 phase maps allowed for mapping of all images
to a common phase, which proved essential for the reconstruction. In future, extension
of the proposed work to other types of pulse sequences is likely to be less phase sensitive.
55
56
4.1.1 ~d and -~d “Checkerboard” Acquisition Scheme
The utilization of both ~d and -~d diffusion directions for imaging has interesting appli-
cations for future use. Currently, most diffusion imaging protocols will only use ~d, and
assume that ~d and -~d are identical – as they should be based on the underlying physical
explanation if diffusion imaging. However, this assumption may fail if scan imperfections
cause other intensity alterations. By acquiring both directions, in the best case scenario,
no differences will be found and the image will be identical to the original method. On
the other hand, the mixing of both provides the opportunity to recognize and correct
erroneous phase, producing a cleaner data set for diffusion modelling. In this study this
method of sampling proved critical.
While this is largely attributable to our use of a RARE readout, the checkerboard
acquisition scheme is not limited to working with RARE acquisitions, nor is it limited
to sampling Cartesian k-space. In theory, all this method requires is a similar number
of ~d and -~d samples such that the phase of each can be defined. Here, the checkerboard
pattern was used because inconsistencies between the datasets were easily identified in
the corners of the image. However, one could imagine an experiment where both ~d and
-~d are probed, but with a spiral trajectory through k-space. On each TR, the sign of
~d could change, but the aliased inconsistencies would spread in a less coherent fashion
around the periphery of the image. A more likely application would be in multi-shot
EPI, with each shot could alternating sign on ~d, and therefore give a mixed dataset that
could help elucidate the differences present between ~d and -~d that should be removed
before the use of any models.
4.2 Limitations of the Study
4.2.1 Phase Measurements
In pursuing my thesis research, it became clear to me that the most important part of
the CS reconstruction for MRI is the phase that is used to move from real to complex
space (and back again). This represents the critical link between image space and the
57
acquired data within the reconstruction. For many proof-of-principle demonstrations
of CS, fully-sampled data is retrospectively undersampled and the phase is taken as a
known quantity [72]. However obtaining an accurate phase for the CS algorithm is more
difficult given undersampled data. For this reason, CS may work best when samples
can be meticulously shimmed and employed with sequences relatively robust to phase
errors. Unfortunately, the RARE sequence employed here is particularly sensitive to
phase-induced artifact. As a means of compensation here, the redundancy of diffusion
MRI, fully sampled b0 images, and the checkerboard pattern of the diffusion directions
were employed to adapt or accommodate phases in each image.
4.2.2 Trade-Offs
Adequate resolution and sampling in all four (or five) of the dimensions in a diffusion MRI
experiment are critical to accurate diffusion modelling and delineation of the structures
of interest. Too great an undersampling in any one of the dimensions will lead to unsatis-
factory image results. In the mouse however, given the extensive length of time required
to image ex vivo brain samples at high resolution, this trade off of spatial information
through undersampling for an increased number of diffusion directions enables advanced
modelling techniques available such as HARDI. This trade-off will inevitably result in
poorer apparent spatial resolution, to the benefit of a much richer mapping of diffusion
direction space. The use of CS in this research made this trade-off feasible. Additional
research will be required in a cohort of mouse models with representative pathologies
in order to further optimize the balance between the two. The selection of 25% spatial
sampling in this research was empirically selected based on the appearance of structure
in wildtype mice only, and did not take into account the appearance of pathology.
4.2.3 Empirical Tuning of the Weighting Factors (λi)
Tuning of the different λ weighting values for any optimization is important, especially in
nonlinear optimization, as a small change in input can lead to a large change in output.
Since λs are empirically chosen, they may need to be retuned depending on the contrast,
58
undersampling, or potentially the SNR of the dataset in use. Hence, the search space
has the potential to be quite large and may require a significant amount of retuning after
adjustment of the application, imaging protocol, or type of input data.
4.3 Future Work
4.3.1 Further Applications in Diffusion
In this body of work, the CS implementation focused on increasing the number of direc-
tions to utilize a more advanced model of diffusion. However, it would be particularly
interesting to consider alternative models of diffusion where the use of only one b-value
is not sufficient. In principle, a very similar approach could be used with a few b-values
with fewer diffusion directions for each. For example, if the same undersampling method
was used, in theory there could be four shells with 30 directions each. However, as the
directions are further apart in this case, the diffusion TV term proposed in this work
is forced to consider a broader solid angle (or be reduced to a simple anatomical CS
reconstruction). The implemented CS algorithm and TVdiff constraint are able to handle
this trade-off as well, though it might be of interest to explore the potential of using
constraints across not only diffusion directions but across different b-values as well.
The current method of increasing the number of diffusion directions, while decreasing
the acquired spatial data for each direction could be taken even further. On one side, one
could imagine trying to acquire fewer directions with greater spatial sampling in each,
for example, 90 directions at 33% sampling each. This would allow each independent
reconstruction to be more reliant on the data collected for that specific direction and less
dependent on sharing information between directions. By constraining the comparison
with a user-defined solid angle, there is the potential to adapt this constraint so that the
CS algorithm is more or less dependent on the comparison across diffusion directions. At
the other extreme, one could imagine taking a continuum of directions with very little
spatial data collected for each one. This would require modifications to the CS algorithm
used here since each individual direction would have too little data to be used in any way
59
on its own. However, such an approach might have the advantage of being user tunable
retrospectively, letting directions be combined or divided according to the analysis needs.
While appealing, this strategy would likely be extremely dependent on accurate handling
of phase, especially since accurate representations of phase estimates require a minimum
of data with a fixed diffusion weighting. Thus in order for this to work, an algorithm
would need to be put in place to estimate the phase correction(s) required for each
direction based on the directions around it. In my experience, the sensitivity of CS to
phase means that this algorithm would need to be exceptionally robust. Advantageously,
the phase corrections required generally vary smoothly with gradient amplitude, so such
an approach is not out of the question. The range of possibilities are even broader if one
adds a wide range of b-values to the continuum of sampled directions
4.3.2 Potential Use in Clinic
Within the clinic, 2D imaging techniques are generally used with trajectories such as
EPI for an image readout, which is very different from 3D imaging with RARE used in
the ex vivo mouse MRI experiments presented here. Even though the faster 2D imaging
methods are in use, HARDI acquisitions still require long scan times, and thus are not in
broad use [103]. However, if the acquisition and reconstruction scheme discussed in this
thesis could be adapted to use a different trajectory, there is potential for it to accelerate
other scans as well. If this avenue is explored, optimization of the sampling scheme, and
the phase estimation are likely to remain key factors in success. Lustig emphasized as
much when he said “randomness is too important to be left to chance” [72].
Chapter 5
Conclusion
This study focused on the potential of using CS for diffusion MRI in the mouse by making
a trade-off between spatial sampling and the number of diffusion directions acquired. This
will enable the use of more sophisticated models of diffusion to characterize mouse models
of human disease. The implementation was based on the idea that nearby diffusion
directions have very similar intensities and may thereby serve as a constraint in the
CS-based reconstruction.
Numerous challenges were encountered in implementation, exclusively related to var-
ious kinds of induced image phase. A novel method where both ±~d were probed for a
given diffusion direction was implemented to accommodate and correct induced phase
errors. While this method may require further testing for use in other applications, the
CS reconstruction after phase correction was found to produce satisfactory image quality
in the spatial dimensions, and a significantly accelerated acquisition of diffusion direction
information. I demonstrated the approach by collection of 120-direction diffusion data
with 4-fold undersampling, for comparison to fully-sampled 30-direction data. The new
approach largely preserved anatomical detail while enabling a 4-fold increase in sampling
of diffusion directions. The new method further enabled the use of higher order models,
impossible for the data with fewer directions. Expansion of this approach will enable use
NODDI, or DKI framework where more than one b-value is also used. As the use of MRI
in preclinical applications continues to expand, the demand for quantitative measures
and an evaluation of their potential to reveal microstructural changes motivates an ex-
60
61
panded evaluation of how spatial, diffusion, and indeed other contrasts may be combined
to maximize information content per unit scan time. CS and model-based analyses have
the potential to play and important role in the forward progress of this work.
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